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A137441 Partial sums of partial sums of PrimePi(k).

%I A137441 #15 Oct 07 2021 20:45:09
%S A137441 0,1,4,9,17,28,43,62,85,112,144,181,224,273,328,389,457,532,615,706,
%T A137441 805,912,1028,1153,1287,1430,1582,1743,1914,2095,2287,2490,2704,2929,
%U A137441 3165,3412,3671,3942,4225,4520,4828,5149,5484,5833,6196,6573,6965,7372,7794
%N A137441 Partial sums of partial sums of PrimePi(k).
%H A137441 Alois P. Heinz, <a href="/A137441/b137441.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harvey P. Dale)
%F A137441 a(n) = partial sum of A046992 = partial sum of partial sum of PrimePi(k) = partial sum of partial sum of A000720(k) = Sum_{j=1..n} (Sum_{k=1..j} PrimePi(k)).
%F A137441 a(n) = Sum_{i=1..n} (n+1-i)*pi(i), where pi = A000720. - _Ridouane Oudra_, Aug 31 2019
%p A137441 A000720 := proc(n) option remember ; numtheory[pi](n) ; end: A046992 := proc(n) option remember ; add( A000720(i),i=1..n) ; end: A137441 := proc(n) add( A046992(i),i=1..n) ; end: seq(A137441(n),n=1..80) ; # _R. J. Mathar_, Apr 23 2008
%p A137441 # second Maple program:
%p A137441 b:= proc(n) option remember; `if`(n<1, [0$2],
%p A137441       (p-> p+[numtheory[pi](n+1), p[1]])(b(n-1)))
%p A137441     end:
%p A137441 a:= n-> b(n)[2]:
%p A137441 seq(a(n), n=1..49);  # _Alois P. Heinz_, Oct 07 2021
%t A137441 Accumulate[Accumulate[PrimePi[Range[50]]]] (* _Harvey P. Dale_, Feb 17 2013 *)
%Y A137441 Cf. A000720, A046992.
%K A137441 easy,nonn
%O A137441 1,3
%A A137441 _Jonathan Vos Post_, Apr 17 2008
%E A137441 More terms from _R. J. Mathar_, Apr 23 2008